Problem 44259. Product of two multivariate polynomials

Created by Andrew Newell

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{"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","targetMode":"","relationshipId":"rId1","target":"/matlab/document.xml"},{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/output","targetMode":"","relationshipId":"rId2","target":"/matlab/output.xml"}],"parts":[{"partUri":"/matlab/document.xml","relationship":[],"contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n<w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>MATLAB</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/help/matlab/polynomials.html\"><w:r><w:t>has a few functions</w:t></w:r></w:hyperlink><w:r><w:t> for creating and manipulating single-variable polynomials, but outside of the Symbolic Math Toolbox there is nothing for multivariate polynomials such as</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>y-x^2</w:t></w:r><w:r><w:t> or</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>x^2+y^2+z^2-1</w:t></w:r><w:r><w:t>. Generalizing the approach for one variable, we can define an array of coefficients with one dimension for each variable. We will index them in decreasing order, for example if</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>p(x) = A + B x^3</w:t></w:r><w:r><w:t>, then the coefficients are</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[c = [B 0 0 A].]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Note this is same order as used by the builtin functions. A couple more examples: if</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>p(x,y) = x - y^2</w:t></w:r><w:r><w:t>, then</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[c = [0 -1; 0 0; 1 0].]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>If</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>p(x,y,z) = z-2</w:t></w:r><w:r><w:t>, then</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>c</w:t></w:r><w:r><w:t> has dimensions</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>[1 1 2]</w:t></w:r><w:r><w:t> with</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>c(:,:,1) = 1</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>c(:,:,2) = -2</w:t></w:r><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>The challenge is to create a function</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>polyMult</w:t></w:r><w:r><w:t> that takes two arrays of coefficients for polynomials</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>p1</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>p2</w:t></w:r><w:r><w:t> and returns the coefficients for</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>p1*p2</w:t></w:r><w:r><w:t>. See the tests for examples.</w:t></w:r></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray type=\"array\"/></embeddedOutputs>"}]}

MATLAB <a href = "https://www.mathworks.com/help/matlab/polynomials.html">has a few functions</a> for creating and manipulating single-variable polynomials, but outside of the Symbolic Math Toolbox there is nothing for multivariate polynomials such as <tt>y-x^2</tt> or <tt>x^2+y^2+z^2-1</tt>. Generalizing the approach for one variable, we can define an array of coefficients with one dimension for each variable. We will index them in decreasing order, for example if <tt>p(x) = A + B x^3</tt>, then the coefficients are<pre class="language-matlab">c = [B 0 0 A].
</pre>Note this is same order as used by the builtin functions. A couple more examples: if <tt>p(x,y) = x - y^2</tt>, then<pre class="language-matlab">c = [0 -1; 0 0; 1 0].
</pre>If <tt>p(x,y,z) = z-2</tt>, then <tt>c</tt> has dimensions <tt>[1 1 2]</tt> with <tt>c(:,:,1) = 1</tt> and <tt>c(:,:,2) = -2</tt>.The challenge is to create a function <tt>polyMult</tt> that takes two arrays of coefficients for polynomials <tt>p1</tt> and <tt>p2</tt> and returns the coefficients for <tt>p1*p2</tt>. See the tests for examples.

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Last Solution submitted on Jun 12, 2022

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Predrag Vukovic on 23 Nov 2020

Can you explain why is the same matrix c = [0 -1; 0 0 ; 1 0] used to represent two different polynomials p(x,y) = x-y^2 (as in problem definition above) and p(x,y) = y - x^2 as in test example 3 on the solution page?

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Problem 44259. Product of two multivariate polynomials

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